domain of y=sqrt(x+7)+sqrt(x-7)

{ "query": { "display": "domain $$y=\\sqrt{x+7}+\\sqrt{x-7}$$", "symbolab_question": "FUNCTION#domain y=\\sqrt{x+7}+\\sqrt{x-7}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x\\ge 7", "interval": "[7,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\sqrt{x+7}+\\sqrt{x-7}\\::{\\quad}x\\ge\\:7$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find non-negative values for radicals:$${\\quad}x\\ge\\:7$$", "input": "\\sqrt{x+7}+\\sqrt{x-7}", "steps": [ { "type": "step", "primary": "$$\\sqrt{f\\left(x\\right)}\\quad\\Rightarrow\\quad\\:f\\left(x\\right)\\ge{0}\\:$$", "meta": { "general_rule": { "extension": "$$\\sqrt{f\\left(x\\right)}$$ (or any even root) has real values only when $$f\\left(x\\right)\\ge\\:0\\:$$" } } }, { "type": "interim", "title": "Solve $$x+7\\ge\\:0:{\\quad}x\\ge\\:-7$$", "input": "x+7\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "x+7\\ge\\:0", "result": "x\\ge\\:-7", "steps": [ { "type": "step", "primary": "Subtract $$7$$ from both sides", "result": "x+7-7\\ge\\:0-7" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:-7" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x-7\\ge\\:0:{\\quad}x\\ge\\:7$$", "input": "x-7\\ge\\:0", "steps": [ { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "x-7\\ge\\:0", "result": "x\\ge\\:7", "steps": [ { "type": "step", "primary": "Add $$7$$ to both sides", "result": "x-7+7\\ge\\:0+7" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:7" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "Combine the intervals", "result": "x\\ge\\:7" } ], "meta": { "interimType": "Non Negative Radicals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6nJ93N20v/1Qi4UC8u/a/1simP38IY5/ctHdL1kqSGEAgTvMMQ//tbxv6+qLGGfm8xPSGwkIAxHXPuFmizxnhQubv/YGT5yt+kuBAQHJL+hAZEt3ZXAiqUE0HIXrrrezJCbu1QylidsvhQn3xyyWzUznpSNNPrXAWrGtmg1+u3776JRVJJBRuNbjWZgQvayhnyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "The function domain", "result": "x\\ge\\:7" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sqrt{x+7}+\\sqrt{x-7}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }